Elastic rod models for natural and synthetic polymers: analytical solutions for arc-length dependent elasticity Silvana De Lillo, Gaia Lupo, Matteo Sommacal.

Elastic rod models for natural and synthetic polymers: analytical Elastic rod models for natural and synthetic polymers: analytical solutions for arc-length dependent elasticitysolutions for arc-length dependent elasticity

Silvana De Lillo, Gaia Lupo, Matteo SommacalSilvana De Lillo, Gaia Lupo, Matteo SommacalDipartimento di Matematica e Informatica and CR-INSTM VILLAGE Università di Perugia Dipartimento di Matematica e Informatica and CR-INSTM VILLAGE Università di Perugia

INFN sezione di PerugiaINFN sezione di Perugia

MMario Argeri,Vincenzo Baroneario Argeri,Vincenzo BaroneDipartimento di Chimica and CR-INSTM VILLAGE, Università Federico II di NapoliDipartimento di Chimica and CR-INSTM VILLAGE, Università Federico II di Napoli

CNR-IPCF PisaCNR-IPCF Pisa

11

22

Properties of different helices

Helix radius

Pitch Res.x turn

Rise x res.

A - DNAB - DNAZ - DNA

1.3 nm1.0 nm0.9 nm

2.46 nm3.32 nm4.56 nm

10.710.412.0

0.23 nm0.33 nm0.38 nm

-helix3/10 helix-helixCollagen

0.23 nm0.19 nm0.28 nm0.16 nm

0.54 nm0.60 nm0.47 nm0.96 nm

3.63.04.33.3

0.15 nm0.20 nm0.11 nm0.29 nm

33

A, B, (right-handed helices) and Z (left-handed helix) forms

of DNA

44

PRION DISEASESPRION DISEASESPRION DISEASESPRION DISEASES

Key eventKey event: : CONFORMATIONAL TRANSITIONCONFORMATIONAL TRANSITION

ClassificationClassification: neurodegenerative diseases: neurodegenerative diseases

Pathogen: Pathogen: PrPPrPScSc

PrPPrPCC

Which is the mechanism underlying conformational transition Which is the mechanism underlying conformational transition

of PrPof PrPCC to PrP to PrPScSc??Which is the mechanism underlying conformational transition Which is the mechanism underlying conformational transition

of PrPof PrPCC to PrP to PrPScSc??

Which factors do enhance the conformational transitionWhich factors do enhance the conformational transition??Which factors do enhance the conformational transitionWhich factors do enhance the conformational transition??

??PrPPrPScSc amyloid-like fibrilsamyloid-like fibrils

--helix 40 %helix 40 % 30 %30 %--sheet 3 %sheet 3 % 43 %43 %

55

Representative conformations of infinite homopolypeptidesRepresentative conformations of infinite homopolypeptides

a

b

c

-sheet(C5)-sheet(C5)

227 7 ribbonribbon

331010 helix (C10) helix (C10)

helixhelix

66

Teflon [a homopolymer (CFTeflon [a homopolymer (CF22))nn ] ]forms 13/6 and 15/7 helicesforms 13/6 and 15/7 helices

c=16.97 c=16.97

77

The numerical model: atomistic The numerical model: atomistic simulationssimulationsGeneral Liquid Optimized Boundary (GLOB) General Liquid Optimized Boundary (GLOB)

G.Brancato, N.Rega, V.Barone, J.Chem.Phys.G.Brancato, N.Rega, V.Barone, J.Chem.Phys. 124, 214505 124, 214505 (2006).(2006).

The external wallThe external wall Constant volumeConstant volume

Bulk reaction fieldBulk reaction field

The Buffer regionThe Buffer region

Bulk densityBulk density

No preferential orientationNo preferential orientation

88

bulkbulk

Nucleosome – DNA complexabout 3x102 DNA bases; 6x104 water molecules: 2x105 atoms

99

1010

The analytical model: elastic stripThe analytical model: elastic stripA.Goriely, M.Tabor, Phys.Rev.Lett. 77,3537-3540 A.Goriely, M.Tabor, Phys.Rev.Lett. 77,3537-3540 (1996)(1996)

In most cases the environment of the helix axis is anisotropic. 1111

The arc length is given by

tdtd

dz

td

dy

td

dxdsts

tt

0

222

0

For an helix we get

tcRtdcR

tdctRtRts

t

t

22

0

22

0

222 cossin

1212

ds

Tdsk

F

)(

The Frenet curvature kkFFss measures the shift from a rectilinear behaviour: it is defined as the modulus of the derivative of the tangent vector w.r.t. the arc length

CurvatureCurvature

2

222

2

p

R

R

cR

R

ds

Tdsk

F

The curvature of a circular The curvature of a circular helix is CONSTANThelix is CONSTANT

1414

The Frenet torsion Fs measures the shift from a planar behaviour

For a circular helix

2

222

2

2

p

R

p

cR

cs

F

The torsion of a circular helixThe torsion of a circular helixis CONSTANTis CONSTANT

1515

x

y

z

O

The strip is characterized bya non null transverse sectionand is subjected to suitabledeformations

Select possible deformations and dynamic variablesSelect possible deformations and dynamic variables Select the forces coming into playSelect the forces coming into play Write the equations associated to static equilibrium configurationsWrite the equations associated to static equilibrium configurations

and determine the geometric shapeof these configurationsand determine the geometric shapeof these configurations 1616

Deformations Deformations (not allowed in our (not allowed in our model)model)

x

y

z

O

Compression,Compression,lengtheninglengthening

shearshear

UndeformedUndeformedconfigurationconfiguration

1717

Deformations Deformations (allowed in our (allowed in our model)model)

x

y

z

O

2 orthogonal bendings2 orthogonal bendings Torsion (twist)Torsion (twist)

UndeformedUndeformedconfigurationconfiguration

1818

KinematicsKinematics

x y

z

O

3d

1d

2d

sr

The elastic strip isdescribed by:

3: ERIr

passing through the centersof the transverse sections

A generalized Frenet frame

sdsdsd 321 ,,

A curve

21,dd Define the plane of the

Transverse section

1919

Dynamic Dynamic variablesvariables

x y

z

O

1d

2d

sr

3d

321 ,, ddd

The frameis orthonormal, so thata vector (Darboux sk

vector) exists that describes the variation of sd

i

3,2,1 isdsksdii

3

1i

iisdksk

21,kk describe the bendingbending

3k describes the twisttwist 2020

x

y

z

O

td

3

2d 1d

b

n

td

bnd

bnd

3

1

1

cossin

sincos

ds

dk

kk

kk

F

F

F

3

2

1

cos

sin

The two frames are The two frames are related by a rotation ofrelated by a rotation ofAngle Angle around around

Describes theintrinsic twistintrinsic twist

sr

3d

2121

ds

dk

kkk

F

F

3

22

21

ForcesForces

x y

z

O

sr

A resulting force A resulting momentum

sF

sM

Internal Efforts Internal Efforts equivalent to

Possibly external forcesexternal forces(gravity, friction) equivalent to

Resulting external force Resulting external momentum

sf

s

sfsF

'

ssFsrsM

'

sr

In general the action of these forcesIn general the action of these forcesdetermines a movement describeddetermines a movement describedby non banal equationsby non banal equations

On the transverse section placed in act:

2424

Eqilibrium equationsEqilibrium equations

x y

z

O

0

0

Fds

rd

ds

Mdds

Fd

sr

In the absence of external forces at equilibrium we get

0

0

3

Fdds

Mdds

Fd

2525

33222111 dbkdkadkasM

Bending stifnessBending stifness Twist Twist stifnessstifness

Jb

EIa

EIa

22

11

22

21

22

21

1

1

4

11

2

22

A

AE

A

A

IJ

IA

J

Rod (with radius A)Rod (with radius A)

Elliptic strip (withElliptic strip (with

semiaxes Asemiaxes A11,A,A22))

E = Young modulus; E = Young modulus; = Shear modulus; = Shear modulus;

II11,I,I22 = principal inertia moments in the cross-section plane = principal inertia moments in the cross-section plane 2626

Equilibrium equations: Equilibrium equations: constitutive relationships constitutive relationships

DKds

Dd

Fdds

Mdds

Fd

ˆˆˆ

0

0

3

0

0

0ˆ

),,(ˆ

12

13

23

321

332211

kk

kk

kk

K

dddD

dFdFdFF

T

2727

212133

13112

22

2

23221

11

1

12213

31132

23321

)(

)(

)(

kkaads

dbk

ds

dkb

Fkkbads

dak

ds

dka

Fkkbads

dak

ds

dka

kFkFds

dF

kFkFds

dF

kFkFds

dF

2828

The Lancret’s theorem

22

22

2

p

R

R

cR

Rk

F2

222

2

2

p

R

p

cR

cF

A helix is a curve, whose tangent makes a constant angle with a fixed lineIn terms of the Frenet frame defined by the so called tangent, normal, andbinormal vectors:

(1)

)()()( sBsNsF

ds

sTdsk

F

)()(

For a general helix For a general helix Lancret’s theorem Lancret’s theorem

states thatstates that

For a circular For a circular helixhelix

),,( BNT

)(

)(

s

sk

F

F

2929

A circular helixcircular helix is describedby the parametric equation

p

ct

tR

tR

t sin

cos

r

R

cp 2

-2-1

0 12

-2-1

012

0

5

10

15

-2-1

012

t

ct

tR

tR

tr sin

cos

R RadiusRadius of the circular cylinder of the circular cylinderalong which the curve is coiledalong which the curve is coiled

c

R

““Speed” of Speed” of advancementadvancementalong the helix axis. along the helix axis. PitchPitch of the helix, i.e. of the helix, i.e. distance between two distance between two successive spires. successive spires.

cp 23030

A. Goriely, M. Nizette, M. Tabor, J. Nonlinear Sci. 11,3-45 (2001)

The (“inverse problem”) approach:- Most of the helices we are interested in are circular

helices (kF and F constant);

- We assign constant values to kF and F ;- We choose the function ;- We solve Kirchhoff’s equations for the six unknowns

F1 , F2 , F3 , a1 , a2 , b

with fixed “initial” values ;

• a1 , a2 , b constant

a1 = a2 (circular rod) leads to arbitrary

a1 ≠ a2 (generic rod) leads to n

• We obtain many new results, in both cases

= and ≡ (s) .

• We recover all the results already present

in literature with a1 , a2 , b constant.

• Energy landscape (variational principle)

• Time evolution

• Two-dimensional limit (ribbon)

Work in progressWork in progress